An interval arithmetic based system and method for solving a global optimization problem is contemplated and provided. Provisions are made for a bisection indexing scheme for a parameter domain of an objective function wherein unique codes are assigned to each iterative interval subset of the parameter domain. Relationships for, between and among iterative subdivisions are arithmetically delimited, with unique codes populating an integer field of a bisection queue system memory component for particular arrays in a bisection context system memory component, and a further integer array of the bisection context. In connection to depth first domain bisection, operations are undertaken relative to the bisection context which are memorialized in relation to the bisection queue, operations which include a work stealing scheme for simultaneous breadth first searching.

An information processing apparatus, program, and information processing method performing validated numerics. Arithmetic operation of definite numbers a.sub.1 and b.sub.1 of the computer numbers in which real numbers A and B are defined by formulas (1) and (2) is performed to determine an absolute effective digit γ satisfying formula (3),
A=a.sub.1+a.sub.2,|a.sub.1|≤C.sup.ha,−C.sup.ea≤a.sub.2<C.sup.ea  formula (1),
B=b.sub.1+b.sub.2,|a.sub.1|≤C.sup.hb,−C.sup.eb≤b.sub.2<C.sup.eb  formula (2),
(A,B)=(a,b).sub.1+(a,b).sub.2,−C.sup.γ≤(a,b).sub.2<C.sup.γ  formula (3).
a.sub.1 and b.sub.1 are definite numbers whose numerical values are definite, and a.sub.2 and b.sub.2 are uncertain numbers whose numerical values are uncertain; C denotes a radix; h.sub.a and h.sub.b denote extended high order maxes that are minimum extended digits satisfying |a.sub.1|≤C.sup.ha and |b.sub.1|≤C.sup.hb, and h.sub.a and h.sub.b denote high order maxes that are integers; and e.sub.a, e.sub.b and γ denote the absolute effective digits that are integers.

Methods for evaluating quantum computing circuits in view of the resource costs of a quantum algorithm are described. A processor-implemented method for performing an evaluation of a polynomial corresponding to an input is provided. The method includes determining a polynomial interpolation for a set of sub-intervals corresponding to the input. The method further includes constructing a quantum circuit for performing, in parallel, polynomial evaluation corresponding to each of the set of sub-intervals.

The apparatus and method for calculating and retaining a bound on error during floating-point operations inserts an additional bounding field into the standard floating-point format that records the retained significant bits of the calculation with notification upon insufficient retention. The bounding field, accounting for both rounding and cancellation errors, includes the lost bits D Field and the accumulated rounding error R Field. The D Field states the number of bits in the floating-point representation that are no longer meaningful. The bounds on the represented real value are determined by the truncated floating-point value and the addition of the error determined by the number of lost bits. The true, real value is absolutely contained by these bounds. The allowable loss (optionally programmable) of significant digits provides a fail-safe, real-time notification of loss of significant digits. This allows representation of real numbers accurate to the last digit.

An information processing apparatus, program, and information processing method performing validated numerics. Arithmetic operation of definite numbers a.sub.1 and b.sub.1 of the computer numbers in which real numbers A and B are defined by formulas (1) and (2) is performed to determine an absolute effective digit γ satisfying formula (3),

A=a.sub.1+a.sub.2, |a.sub.1|≤C.sup.ha, −C.sup.ea≤a.sub.2<C.sup.ea  formula (1),

B=b.sub.1+b.sub.2, |a.sub.1|≤C.sup.hb, −C.sup.eb≤b.sub.2<C.sup.eb  formula (2),

(A,B)=(a,b).sub.1+(a,b).sub.2, −C.sup.γ≤(a,b).sub.2<C.sup.γ  formula (3).

a.sub.1 and b.sub.1 are definite numbers whose numerical values are definite, and a.sub.2 and b.sub.2 are uncertain numbers whose numerical values are uncertain; C denotes a radix; h.sub.a and h.sub.b denote extended high order maxes that are minimum extended digits satisfying |a.sub.1|≤C.sup.ha and |b.sub.1|≤C.sup.hb, and h.sub.a and h.sub.b denote high order maxes that are integers; and e.sub.a, e.sub.b and γ denote the absolute effective digits that are integers.

The apparatus and method for calculating and retaining a bound on error during floating-point operations inserts an additional bounding field into the standard floating-point format that records the retained significant bits of the calculation with notification upon insufficient retention. The bounding field, accounting for both rounding and cancellation errors, includes the lost bits D Field and the accumulated rounding error R Field. The D Field states the number of bits in the floating-point representation that are no longer meaningful. The bounds on the represented real value are determined by the truncated floating-point value and the addition of the error determined by the number of lost bits. The true, real value is absolutely contained by these bounds. The allowable loss (optionally programmable) of significant digits provides a fail-safe, real-time notification of loss of significant digits. This allows representation of real numbers accurate to the last digit.

Apparatus performs various modal interval computations, while accounting for various modal interval operand configurations that are not amenable to ordinary computational operations. Upon detecting an exponent field of all 1's, the apparatus adapts various conventions involving leading bits in the fraction field of the modal interval endpoints to return a result having a useful meaning. Unary, binary and ternary modal interval operations with decorations are contemplated.

The apparatus and method for calculating and retaining a bound on error during floating-point operations inserts an additional bounding field into the standard floating-point format that records the retained significant bits of the calculation with notification upon insufficient retention. The bounding field, accounting for both rounding and cancellation errors, includes the lost bits D Field and the accumulated rounding error R Field. The D Field states the number of bits in the floating-point representation that are no longer meaningful. The bounds on the represented real value are determined by the truncated floating-point value and the addition of the error determined by the number of lost bits. The true, real value is absolutely contained by these bounds. The allowable loss (optionally programmable) of significant digits provides a fail-safe, real-time notification of loss of significant digits. This allows representation of real numbers accurate to the last digit.

The apparatus and method for calculating and retaining a bound on error during floating-point operations inserts an additional bounding field into the standard floating-point format that records the retained significant bits of the calculation with notification upon insufficient retention. The bounding field, accounting for both rounding and cancellation errors, includes the lost bits D Field and the accumulated rounding error R Field. The D Field states the number of bits in the floating-point representation that are no longer meaningful. The bounds on the represented real value are determined by the truncated floating-point value and the addition of the error determined by the number of lost bits. The true, real value is absolutely contained by these bounds. The allowable loss (optionally programmable) of significant digits provides a fail-safe, real-time notification of loss of significant digits. This allows representation of real numbers accurate to the last digit.

Methods for evaluating quantum computing circuits in view of the resource costs of a quantum algorithm are described. A processor-implemented method for performing an evaluation of a polynomial corresponding to an input is provided. The method includes determining a polynomial interpolation for a set of sub-intervals corresponding to the input. The method further includes constructing a quantum circuit for performing, in parallel, polynomial evaluation corresponding to each of the set of sub-intervals.